In this chapter we describe the theory that is behind the algorithms used by Wedderga.
All the rings considered in this chapter are associative and have an identity.
We use the following notation: \(ℚ\) denotes the field of rationals and \(\mathbb F_q\) the finite field of order \(q\). For every positive integer \(k\), we denote a complex \(k\)-th primitive root of unity by \(\xi_k\) and so \(ℚ(\xi_k)\) is the \(k\)-th cyclotomic extension of \(ℚ\).
Given a group \(G\) and a ring \(R\), the group ring \(RG\) over the group \(G\) with coefficients in \(R\) is the ring whose underlying additive group is a right \(R-\)module with basis \(G\) such that the product is defined by the following rule
\[ (gr)(hs)=(gh)(rs) \]
for \(r,s \in R\) and \(g, h \in G\), and extended to \(RG\) by linearity.
A group algebra is a group ring in which the coefficient ring is a field.
We say that a ring \(R\) is semisimple if it is a direct sum of simple left (alternatively right) ideals or equivalently if \(R\) is isomorphic to a direct product of simple algebras each one isomorphic to a matrix ring over a division ring.
By Maschke's Theorem, if \(G\) is a finite group then the group algebra \(FG\) is semisimple if and only the characteristic of the coefficient field \(F\) does not divide the order of \(G\).
In fact, an arbitrary group ring \(RG\) is semisimple if and only if the coefficient ring \(R\) is semisimple, the group \(G\) is finite and the order of \(G\) is invertible in \(R\).
Some authors use the notion semisimple ring for rings with zero Jacobson radical. To avoid confusion we usually refer to semisimple rings as semisimple artinian rings.
If \(R\) is a semisimple ring (9.2) then the Wedderburn decomposition of \(R\) is the decomposition of \(R\) as a direct product of simple algebras. The factors of this Wedderburn decomposition are called Wedderburn components of \(R\). Each Wedderburn component of \(R\) is of the form \(Re\) for \(e\) a primitive central idempotent (9.4) of \(R\).
Let \(FG\) be a semisimple group algebra (9.2). If \(F\) has positive characteristic, then the Wedderburn components of \(FG\) are matrix algebras over finite extensions of \(F\). If \(F\) has zero characteristic then by the Brauer-Witt Theorem [Yam74], the Wedderburn components of \(FG\) are Brauer equivalent (9.5) to cyclotomic algebras (9.11).
The main functions of Wedderga compute the Wedderburn components of a semisimple group algebra \(FG\), such that the coefficient field is either an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals) or a finite field. In the finite case, the Wedderburn components are matrix algebras over finite fields and so can be described by the size of the matrices and the size of the finite field.
In the zero characteristic case each Wedderburn component \(A\) is Brauer equivalent (9.5) to a cyclotomic algebra (9.11) and therefore \(A\) is a (possibly fractional) matrix algebra over cyclotomic algebra and can be described numerically in one of the following three forms:
\[ [n,K], \]
\[ [n,K,k,[d,\alpha,\beta]], \]
\[ [n,K,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le n} ], \]
where \(n\) is the matrix size, \(K\) is the centre of \(A\) (a finite field extension of \(F\)) and the remaining data are integers whose interpretation is explained in 9.12.
In some cases (for the zero characteristic coefficient field) the size \(n\) of the matrix algebras is not a positive integer but a positive rational number. This is a consequence of the fact that the Brauer-Witt Theorem [Yam74] only ensures that each Wedderburn component (9.3) of a semisimple group algebra is Brauer equivalent (9.5) to a cyclotomic algebra (9.11), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component \(D\) of a group algebra can be a division algebra but not a cyclotomic algebra. In this case \(M_n(D)\) is a cyclotomic algebra \(C\) for some \(n\) and therefore \(D\) can be described as \(M_{1/n}(C)\) (see last Example in WedderburnDecomposition (2.1-1)).
The main algorithm of Wedderga is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu [Olt07] which uses previous work by Olivieri, del Río and Simón [OdRS04] (see also [OdR03] ) for rational group algebras of strongly monomial groups (9.17). The algorithms are also based upon the work of Bakshi and Maheshwary [BM14] (see also [BM16]) on the rational group algebras of normally monomial groups (9.18).
A primitive central idempotent of a ring \(R\) is a non-zero central idempotent \(e\) which cannot be written as the sum of two non-zero central idempotents of \(Re\), or equivalently, such that \(Re\) is indecomposable as a direct product of two non-trivial two-sided ideals.
The Wedderburn components (9.3) of a semisimple ring \(R\) are the rings of the form \(Re\) for \(e\) running over the set of primitive central idempotents of \(R\).
Let \(FG\) be a semisimple group algebra (9.2) and \(\chi\) an irreducible character of \(G\) (in an algebraic closure of \(F\)). Then there is a unique Wedderburn component \(A=A_F(\chi)\) of \(FG\) such that \(\chi(A)\ne 0\). Let \(e_F(\chi)\) denote the unique primitive central idempotent of \(FG\) in \(A_F(\chi)\), that is the identity of \(A_F(\chi)\), i.e.
\[ A_F(\chi)=FGe_F(\chi). \]
The centre of \(A_F(\chi)\) is \(F(\chi)=F(\chi(g):g \in G)\), the field of character values of \(\chi\) over \(F\).
The map \(\chi \mapsto A_F(\chi)\) defines a surjective map from the set of irreducible characters of \(G\) (in an algebraic closure of \(F\)) onto the set of Wedderburn components of \(FG\).
Equivalently, the map \(\chi \mapsto e_F(\chi)\) defines a surjective map from the set of irreducible characters of \(G\) (in an algebraic closure of \(F\)) onto the set of primitive central idempontents of \(FG\).
If the irreducible character \(\chi\) of \(G\) takes values in \(F\) then
\[ e_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1}) g. \]
In general one has
\[ e_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi). \]
Let \(K\) be a field. A central simple \(K\)-algebra is a finite dimensional \(K\)-algebra with center \(K\) which has no non-trivial proper ideals. Every central simple \(K\)-algebra is isomorphic to a matrix algebra \(M_n(D)\) where \(D\) is a division algebra (which is finite-dimensional over \(K\) and has centre \(K\)). The division algebra \(D\) is unique up to \(K\)-isomorphisms.
Two central simple \(K\)-algebras \(A\) and \(B\) are said to be Brauer equivalent, or simply equivalent, if there is a division algebra \(D\) and two positive integers \(m\) and \(n\) such that \(A\) is isomorphic to \(M_m(D)\) and \(B\) is isomorphic to \(M_n(D)\).
Let \(R\) be a ring and \(G\) a group.
Intrinsic definition. A crossed product [Pas89] of \(G\) over \(R\) (or with coefficients in \(R\)) is a ring \(R*G\) with a decomposition into a direct sum of additive subgroups
\[ R*G = \bigoplus_{g \in G} A_g \]
such that for each \(g,h\) in \(G\) one has:
* \(A_1=R\) (here \(1\) denotes the identity of \(G\)),
* \(A_g A_h = A_{gh}\) and
* \(A_g\) has a unit of \(R*G\).
Extrinsic definition. Let \(Aut(R)\) denote the group of automorphisms of \(R\) and let \(R^*\) denote the group of units of \(R\).
Let \(a:G \rightarrow Aut(R)\) and \(t:G \times G \rightarrow R^*\) be mappings satisfying the following conditions for every \(g\), \(h\) and \(k\) in \(G\):
(1) \(a(gh)^{-1} a(g) a(h)\) is the inner automorphism of \(R\) induced by \(t(g,h)\) (i.e. the automorphism \(x\mapsto t(g,h)^{-1} x t(g,h)\)) and
(2) \(t(gh,k) t(g,h)^k = t(g,hk) t(h,k)\), where for \(g \in G\) and \(x \in R\) we denote \(a(g)(x)\) by \(x^g\).
The crossed product [Pas89] of \(G\) over \(R\) (or with coefficients in \(R\)), action \(a\) and twisting \(t\) is the ring
\[ R*_a^t G = \bigoplus_{g\in G} u_g R \]
where \(\{u_g : g\in G \}\) is a set of symbols in one-to-one correspondence with \(G\), with addition and multiplication defined by
\[ (u_g r) + (u_g s) = u_g(r+s), \quad (u_g r)(u_h s) = u_{gh} t(g,h) r^h s \]
for \(g,h \in G\) and \(r,s\in R\), and extended to \(R*_a^t G\) by linearity.
The associativity of the product defined is a consequence of conditions (1) and (2) [Pas89].
Equivalence of the two definitions. Obviously the crossed product of \(G\) over \(R\) defined using the extrinsic definition is a crossed product of \(G\) over \(u_1 R\) in the sense of the first definition. Moreover, there is \(r_0\) in \(R^*\) such that \(u_1r_0\) is the identity of \(R*_a^t G\) and the map \(r \mapsto u_1 r_0 r \) is a ring isomorphism \(R \rightarrow u_1R \).
Conversely, let \(R*G=\bigoplus_{g\in G} A_g\) be an (intrinsic) crossed product and select for each \(g\in G\) a unit \(u_g\in A_g\) of \(R*G\). This is called a basis of units for the crossed product \(R*G\). Then the maps \(a:G \rightarrow Aut(R)\) and \(t:G\times G \rightarrow R^*\) given by
\[ r^g = u_g^{-1} r u_g, \quad t(g,h) = u_{gh}^{-1} u_g u_h \quad (g,h \in G, r \in R) \]
satisfy conditions (1) and (2) and \(R*G = R*_a^t G\).
The choice of a basis of units \(u_g \in A_g\) determines the action \(a\) and twisting \(t\). If \(\{u_g \in A_g : g \in G \}\) and \(\{v_g \in A_g : g \in G \}\) are two sets of units of \(R*G\) then \(v_g = u_g r_g\) for some units \(r_g\) of \(R\). Changing the basis of units results in a change of the action and the twisting and so changes the extrinsic definition of the crossed product but it does not change the intrinsic crossed product.
It is customary to select \(u_1=1\). In that case \(a(1)\) is the identity map of \(R\) and \(t(1,g)=t(g,1)=1\) for each \(g\) in \(G\).
Let \(R*G=\bigoplus_{g \in G} A_g\) be a crossed product (9.6) and assume that \(G = \langle g \rangle \) is cyclic. Then the crossed product can be given using a particularly nice description.
Select a unit \(u\) in \(A_{g}\), and let \(a\) be the automorphism of \(R\) given by \(r^a = u^{-1} r u\).
If \(G\) is infinite then set \(u_{g^k} = u^k\) for every integer \(k\). Then
\[ R*G = R[ u | ru = u r^a ], \]
a skew polynomial ring. Therefore in this case \(R*G\) is determined by
\[ [ R, a ]. \]
If \(G\) is finite of order \(d\) then set \(u_{g^k} = u^k\) for \(0 \le k < d\). Then \( b = u^d \in R \) and
\[ R*G = R[ u | ru = u r^a, u^d = b ] \]
Therefore, \(R*G\) is completely determined by the following data:
\[ [ R , [ d , a , b ] ] \]
Let \(R*G=\bigoplus_{g \in G} A_g\) be a crossed product (9.6) and assume that \(G\) is abelian. Then the crossed product can be given using a simple description.
Express \(G\) as a direct sum of cyclic groups:
\[ G = \langle g_1 \rangle \times \cdots \times \langle g_n \rangle \]
and for each \(i=1,\dots,n\) select a unit \(u_i\) in \(A_{g_i}\).
Each element \(g\) of \(G\) has a unique expression
\[ g = g_1^{k_1} \cdots g_n^{k_n}, \]
where \(k_i\) is an arbitrary integer, if \(g_i\) has infinite order, and \(0 \le k_i < d_i\), if \(g_i\) has finite order \(d_i\). Then one selects a basis for the crossed product by taking
\[ u_g = u_{g_1^{k_1} \cdots g_n^{k_n}} = u_1^{k_1} \cdots u_n^{k_n}. \]
* For each \(i=1,\dots, n\), let \(a_i\) be the automorphism of \(R\) given by \(r^{a_i} = u_i^{-1} r u_i\).
* For each \(1 \le i < j \le n\), let \(t_{i,j} = u_j^{-1} u_i^{-1} u_j u_i \in R\).
* If \(g_i\) has finite order \(d_i\), let \(b_i=u_i^{d_i} \in R\).
Then
\[ R*G = R[u_1,\dots,u_n | ru_i = u_i r^{a_i}, u_j u_i = t_{ij} u_i u_j, u_i^{d_i} = b_i (1 \le i < j \le n) ], \]
where the last relation vanishes if \(g_i\) has infinite order.
Therefore \(R*G\) is completely determined by the following data:
\[ [ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i < j \le n} ]. \]
A classical crossed product is a crossed product \(L*_a^t G\), where \(L/K\) is a finite Galois extension, \(G=Gal(L/K)\) is the Galois group of \(L/K\) and \(a\) is the natural action of \(G\) on \(L\). Then \(t\) is a \(2\)-cocycle and the crossed product (9.6) \(L*_a^t G\) is denoted by \((L/K,t)\). The crossed product \((L/K,t)\) is known to be a central simple \(K\)-algebra [Rei03].
A cyclic algebra is a classical crossed product (9.9) \((L/K,t)\) where \(L/K\) is a finite cyclic field extension. The cyclic algebras have a very simple form.
Assume that \(Gal(L/K)\) is generated by \(g\) and has order \(d\). Let \(u=u_g\) be the basis unit (9.6) of the crossed product corresponding to \(g\) and take the remaining basis units for the crossed product by setting \(u_{g^i} = u^i\), (\( i = 0, 1, \dots, d-1 \)). Then \(a = u^n \in K\). The cyclic algebra is usually denoted by \((L/K,a)\) and one has the following description of \((L/K,t)\)
\[ (L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ]. \]
A cyclotomic algebra over \(F\) is a classical crossed product (9.9) \((F(\xi)/F,t)\), where \(F\) is a field, \(\xi\) is a root of unity in an extension of \(F\) and \(t(g,h)\) is a root of unity for every \(g\) and \(h\) in \(Gal(F(\xi)/F)\).
The Brauer-Witt Theorem [Yam74] asserts that every Wedderburn component (9.3) of a group algebra is Brauer equivalent (9.5) (over its centre) to a cyclotomic algebra.
Let \(A=(F(\xi)/F,t)\) be a cyclotomic algebra (9.11), where \(\xi=\xi_k\) is a \(k\)-th root of unity. Then the Galois group \(G=Gal(F(\xi)/F)\) is abelian and therefore one can obtain a simplified form for the description of cyclotomic algebras as for any abelian crossed product (9.8).
Then the \(n \times n\) matrix algebra \(M_n(A)\) can be described numerically in one of the following forms:
* If \(F(\xi)=F\), (i.e. \(G=1\)) then \(A=M_n(F)\) and thus the only data needed to describe \(A\) are the matrix size \(n\) and the field \(F\):
\[ [n,F] \]
* If \(G\) is cyclic (but not trivial) of order \(d\) then \(A\) is a cyclic cyclotomic algebra
\[ A = F(\xi) [ u | \xi u = u \xi^\alpha, u^d = \xi^\beta ] \]
and so \(M_n(A)\) can be described with the following data
\[ [n,F,k,[d,\alpha,\beta]], \]
where the integers \(k\), \(d\), \(\alpha\) and \(\beta\) satisfy the following conditions:
\[ \alpha^d \equiv 1 \; mod \; k, \quad \beta(\alpha-1) \equiv 0 \; mod \; k. \]
* If \(G\) is abelian but not cyclic then \(M_n(A)\) can be described with the following data (see 9.8):
\[ [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ] \]
representing the \(n \times n\) matrix ring over the following algebra:
\[ A = F(\xi)[ u_1, \ldots, u_m \mid \xi u_i = u_i \xi^{\alpha_i}, \quad u_i^{d_i}=\xi^{\beta_i}, \quad u_s u_r = \xi^{\gamma_{rs}} u_r u_s, \quad i = 1, \ldots, m, \quad 0 \le r < s \le m ] \]
where
* \(\{g_1,\ldots,g_m\}\) is an independent set of generators of \(G\),
* \(d_i\) is the order of \(g_i\),
* \(\alpha_i\), \(\beta_i\) and \(\gamma_{rs}\) are integers, and
\[ \xi^{g_i} = \xi^{\alpha_i}. \]
Let \(G\) be a finite group and \(F\) a field whose characteristic does not divide the order of \(G\). If \(H\) is a subgroup of \(G\) then set
\[ \widehat{H} = |H|^{-1}\sum_{x \in H} x. \]
The element \(\widehat{H}\) is an idempotent of \(FG\) which is central in \(FG\) if and only if \(H\) is normal in \(G\).
If \(H\) is a proper normal subgroup of a subgroup \(K\) of \(G\) then set
\[ \varepsilon(K,H) = \prod_{L} (\widehat{N}-\widehat{L}) \]
where \(L\) runs on the normal subgroups of \(K\) which are minimal among the normal subgroups of \(K\) containing \(N\) properly. By convention, \(\varepsilon(K,K)=\widehat{K}\). The element \(\varepsilon(K,H)\) is an idempotent of \(FG\).
If \(H\) and \(K\) are subgroups of \(G\) such that \(H\) is normal in \(K\) then \(e(G,K,H)\) denotes the sum of all different \(G\)-conjugates of \(\varepsilon(K,H)\). The element \(e(G,K,H)\) is central in \(FG\). In general it is not an idempotent but if the different conjugates of \(\varepsilon(K,H)\) are orthogonal then \(e(G,K,H)\) is a central idempotent of \(FG\).
If \((K,H)\) is a Shoda Pair (9.14) of \(G\) then there is a non-zero rational number \(a\) such that \(ae(G,K,H))\) is a primitive central idempotent (9.4) of the rational group algebra \(ℚ G\). If \((K,H)\) is a strong Shoda pair (9.15) of \(G\) then \(e(G,K,H)\) is a primitive central idempotent of \(ℚ G\).
Assume now that \(F\) is a finite field of order \(q\), \((K,H)\) is a strong Shoda pair of \(G\) and \(C\) is a cyclotomic class of \(K/H\) containing a generator of \(K/H\). Then \(e_C(G,K,H)\) is a primitive central idempotent of \(FG\) (see 9.19).
Let \(G\) be a finite group. A Shoda pair of \(G\) is a pair \((K,H)\) of subgroups of \(G\) for which there is a linear character \(\chi\) of \(K\) with kernel \(H\) such that the induced character \(\chi^G\) in \(G\) is irreducible. By [Sho33] or [OdRS04], \((K,H)\) is a Shoda pair if and only if the following conditions hold:
* \(H\) is normal in \(K\),
* \(K/H\) is cyclic and
* if \(K^g \cap K \subseteq H\) for some \(g \in G\) then \(g \in K\).
If \((K,H)\) is a Shoda pair and \(\chi\) is a linear character of \(K\le G\) with kernel \(H\) then the primitive central idempotent (9.4) of \(ℚ G\) associated to the irreducible character \(\chi^G\) is of the form \(e=e_ℚ (\chi^G)=a e(G,K,H)\) for some \(a \in ℚ \) [OdRS04] (see 9.13 for the definition of \(e(G,K,H)\)). In that case we say that \(e\) is the primitive central idempotent realized by the Shoda pair \((K,H)\) of \(G\).
A group \(G\) is monomial, that is every irreducible character of \(G\) is monomial, if and only if every primitive central idempotent of \(ℚ G\) is realizable by a Shoda pair of \(G\).
A strong Shoda pair of \(G\) is a pair \((K,H)\) of subgroups of \(G\) satisfying the following conditions:
* \(H\) is normal in \(K\) and \(K\) is normal in the normalizer \(N\) of \(H\) in \(G\),
* \(K/H\) is cyclic and a maximal abelian subgroup of \(N/H\) and
* for every \(g \in G\setminus N\) , \(\varepsilon(K,H)\varepsilon(K,H)^g=0\). (See 9.13 for the definition of \(\varepsilon(K,H)\)).
Let \((K,H)\) be a strong Shoda pair of \(G\). Then \((K,H)\) is a Shoda pair (9.14) of \(G\). Thus there is a linear character \(\theta\) of \(K\) with kernel \(H\) such that the induced character \(\chi=\chi(G,K,H)=\theta^G\) is irreducible. Moreover the primitive central idempotent (9.4) \(e_{ℚ }(\chi)\) of \(ℚ G\) realized by \((K,H)\) is \(e(G,K,H)\), see [OdRS04].
Two strong Shoda pairs (9.15) \((K_1,H_1)\) and \((K_2,H_2)\) of \(G\) are said to be equivalent if the characters \(\chi(G,K_1,H_1)\) and \(\chi(G,K_2,H_2)\) are Galois conjugate, or equivalently if \(e(G,K_1,H_1)=e(G,K_2,H_2)\). A set of representatives of strong Shoda pairs of \(G\) is termed as a complete irredundant set of strong Shoda pairs of \(G\).
The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra \(FGe_F(\chi)\) as a matrix algebra of a cyclotomic algebra (9.11, see [OdRS04] for \(F=ℚ \) and [Olt07] for the general case).
More precisely, \(ℚ Ge(G,K,H)\) is isomorphic to \(M_n(ℚ (\xi)*_a^t N/K)\), where \(\xi\) is a \([K:H]\)-th root of unity, \(N\) is the normalizer of \(H\) in \(G\), \(n=[G:N]\) and \(ℚ (\xi)*_a^t N/K\) is a crossed product (see 9.6) with action \(a\) and twisting \(t\) given as follows:
Let \(x\) be a fixed generator of \(K/H\) and \(\varphi : N/K \rightarrow N/H\) a fixed left inverse of the canonical projection \(N/H\rightarrow N/K\). Then
\[ \xi^{a(r)} = \xi^i, \mbox{ if } x^{\varphi(r)}= x^i \]
and
\[ t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j, \]
for \(r,s \in N/K\) and integers \(i\) and \(j\), see [OdRS04]. Notice that the cocycle is the one given by the natural extension
\[ 1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1 \]
where \(K/H\) is identified with the multiplicative group generated by \(\xi\). Furthermore the centre of the algebra is \(ℚ (\chi)\), the field of character values over \(ℚ \), and \(N/K\) is isomorphic to \(Gal(ℚ (\xi)/ℚ (\chi))\).
If the rational field is changed to an arbitrary ring \(F\) of characteristic \(0\) then the Wedderburn component \(A_F(\chi)\), where \(\chi = \chi(G,K,H)\) is isomorphic to \(F(\chi)\otimes_{ℚ (\chi)}A_ℚ (\chi)\). Using the description given above of \(A_ℚ (\chi)=ℚ G e(G,K,H)\) one can easily describe \(A_F(\chi)\) as \(M_{nd}(F(\xi)/F(\chi),t')\), where \(d=[ℚ (\xi): ℚ(\chi)]/[F(\xi):F(\chi)]\) and \(t'\) is the restriction to \(Gal(F(\xi)/F(\chi))\) of \(t\) (a cocycle of \(N/K = Gal(ℚ (\xi)/ℚ (\chi))\)).
An extremely strong Shoda pair of \(G\) is a pair \((K,H)\) of subgroups of \(G\) satisfying the following conditions:
* \(K\) is normal in \(G\),
* \(K/H\) is cyclic and a maximal abelian subgroup of \(N/H\), where \(N\) is the normalizer of \(H\) in \(G\).
Let \((K,H)\) be an extremely strong Shoda pair of \(G\). Then \((K,H)\) is a strong Shoda pair (9.15) of \(G\), with \(K\) normal in \(G\) [BM14], so that there is a linear character \(\theta\) of \(K\) with kernel \(H\) such that the induced character \(\chi=\chi(G,K,H)=\theta^G\) is irreducible. Moreover, the primitive central idempotent \(e_{ℚ }(\chi)\) of \(ℚ G\) realized by \((K,H)\) is \(e(G,K,H)\) (9.4) and one can describe the associated simple algebra (9.15). Two extremely strong Shoda pairs of \(G\) are said to be equivalent if they are equivalent as strong Shoda pairs (9.15). A set of representatives of extremely strong Shoda pairs of \(G\) is called a complete irredundant set of extremely strong Shoda pairs of \(G\) [BM14].
If \(G\) is a normally monomial group (9.18), then the set of primitive central idempotents of the rational group algebra realized by strong Shoda pairs of \(G\) is same as the one realized by extremely strong Shoda pairs of \(G\) [BM14]. The algorithm to compute a complete irredundant set of extremely strong Shoda pairs of \(G\) has been explained in [BM16].
Let \(G\) be a finite group an \(\chi\) an irreducible character of \(G\).
One says that \(\chi\) is strongly monomial if there is a strong Shoda pair (9.15) \((K,H)\) of \(G\) and a linear character \(\theta\) of \(K\) of \(G\) with kernel \(H\) such that \(\chi=\theta^G\).
The group \(G\) is strongly monomial if every irreducible character of \(G\) is strongly monomial.
Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial [OdRS04]. The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in [OdR03]. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in [BdR07] (see Section 9.19). Finally, Olteanu [Olt07] shows how to compute the Wedderburn decomposition (9.3) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of \(G\) but also the strong Shoda pairs of the subgroups of \(G\).
Let \(G\) be a finite group and \(\chi\) be an irreducible character of \(G\).
One says that \(\chi\) is normally monomial if there is a normal subgroup \(K\) of \(G\) such that \(\chi\) is induced from a linear character of \(K\).
The group \(G\) is normally monomial if every irreducible character of \(G\) is normally monomial. Bakshi and Maheshwary proved that if \(G\) is a normally monomial group, then for every irreducible character \(\chi\) of \(G\), there exists an extremely strong Shoda pair \((K,H)\) of \(G\) (9.16) such that \(\chi=\theta^G\), where \(\theta\) is a linear character of \(K\) with kernel \(H\) [BM14].
.
Let \(G\) be a finite group and \(F\) a finite field of order \(q\), coprime to the order of \(G\).
Given a positive integer \(n\), coprime to \(q\), the \(q\)-cyclotomic classes modulo \(n\) are the set of residue classes module \(n\) of the form
\[ \{i,iq,iq^2,iq^3, \dots \} \]
The \(q\)-cyclotomic classes module \(n\) form a partition of the set of residue classes module \(n\).
A generating cyclotomic class module \(n\) is a cyclotomic class containing a generator of the additive group of residue classes module \(n\), or equivalently formed by integers coprime to \(n\).
Let \((K,H)\) be a strong Shoda pair (9.15) of \(G\) and set \(n=[K:H]\). Fix a primitive \(n\)-th root of unity \(\xi\) in some extension of \(F\) and an element \(g\) of \(K\) such that \(gH\) is a generator of \(K/H\). Let \(C\) be a generating \(q\)-cyclotomic class modulo \(n\). Then set
\[ \varepsilon_C(K,H) = [K:H]^{-1} \widehat{H} \sum_{i=0}^{n-1} tr(\xi^{-ci})g^i, \]
where \(c\) is an arbitrary element of \(C\) and \(tr\) is the trace map of the field extension \(F(\xi)/F\). Then \(\varepsilon_C(K,H)\) does not depend on the choice of \(c \in C\) and is a primitive central idempotent (9.4) of \(FK\).
Finally, let \(e_C(G,K,H)\) denote the sum of the different \(G\)-conjugates of \(\varepsilon_C(K,H)\). Then \(e_C(G,K,H)\) is a primitive central idempotent (9.4) of \(FG\) [BdR07]. We say that \(e_C(G,K,H)\) is the primitive central idempotent realized by the strong Shoda pair \((K,H)\) of the group \(G\) and the cyclotomic class \(C\).
If \(G\) is strongly monomial (9.17) then every primitive central idempotent of \(FG\) is realizable by some strong Shoda pair (9.15) of \(G\) and some cyclotomic class \(C\) [BdR07]. As in the zero characteristic case, this explain how to compute the Wedderburn decomposition (9.3) of \(FG\) for a finite semisimple algebra of a strongly monomial group (see [BdR07] for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.
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(By Allen Herman, May 2013. Updated October 2014.)
The division algebra parts of simple algebras in the Wedderburn Decomposition of the group algebra of a finite group over an abelian number field \(F\) correspond to elements of the Schur Subgroup \(S(F)\) of the Brauer group of \(F\). Like all classes in the Brauer group of an algebraic number field \(F\), the division algebra part of a representative of a given Brauer class is determined up to \(F\)-algebra isomorphism by its list of local Hasse invariants at all primes (i.e. places) of \(F\). The local invariant at a prime \(P\) of \(F\) is a lowest terms fraction \(r/m_P\) whose denominator is the local Schur index \(m_P\) of the simple algebra at the prime \(q\) (see [Rei03]). For division algebras whose Brauer class lies in the Schur Subgroup of an abelian number field \(F\), the local indices at any of the primes \(P\) lying over the same rational prime \(p\) are equal to the same positive integer \(m_p\), and the numerator of the local invariants among these primes are uniformly distributed among the integers \(r\) coprime to \(m_p\) [BS72].
The local Schur index functions in wedderga produce a list of the nontrivial local indices of the division algebra part of the simple algebra at all rational primes. The Schur index of the simple algebra over \(F\) is the least common multiple \(m\) of these local indices, and the dimension of the division algebra part of the simple algebra over \(F\) is \(m^2\). While not sufficient to identify these division algebras up to ring isomorphism in general, this list of local indices does identify the division algebra up to ring isomorphism whenever there is no pair of local indices at odd primes that are greater than 2. (This is at least the case for groups of order less than 3^2*7*13.) So it gives the information desired in most basic situations, and allows one to distinguish almost all pairs of simple components of group algebras.
Wedderga's functions compute local indices for generalized quaternion algebras defined over the rationals and cyclotomic algebras defined over any abelian number field. Special shortcut functions are available for cyclic cyclotomic algebras. There are also versions of the functions that compute the local and global Schur index of a character of a finite group over a given abelian number field. The steps in the general character- theoretic method involve 1) a Brauer-Witt reduction to a cyclic-by-abelian group, 2) use of the Frobenius-Schur indicator to compute the local index at infinity, 3) computing the \(p\)-local index for an ordinary irreducible character \(\chi\) of a \(p\)-solvable group using the values of an irreducible Brauer character in the same \(p\)-block in cases where the \(p\)-defect group of \(\chi\) is cyclic, and 4) use of Riese and Schmid's characterization of dyadic Schur groups ([Sch94] and [RS96]) to handle the exceptional cases where step 3) is not available. Our approach to rational quaternion algebras is the standard one given, for example, in [Pie82]. The Legendre symbol operation in GAP is used to determine the local index at odd primes. The local index of the generalized quaternion algebra \((a,b)\) over \(Q\) at the infinite prime will be \(2\) if both \(a\) and \(b\) are negative, and otherwise \(1\). We avoid the complicated case of quadratic reciprocity when working over Q by using the Hasse-Brauer-Albert-Noether Theorem ([Rei03], pg. 276): since we know the other primes of \(Q\) where the local index is \(2\), it determines the local index at the prime \(2\). For generalized quaternion algebras over number fields \(F\) other than \(Q\), we have to convert to cyclic or cyclic cyclotomic algebras and use the other local index functions, or appeal to a number theory system outside of GAP that can solve norm equations.
There are three shortcut functions used to compute local indices of cyclic cyclotomic algebras, which wedderga's -Info functions produce in the form \([r,F,n,[a,b,c]]\). The local index at infinity is calculated by determining if the real completion of the corresponding algebra will produce a real quaternion algebra. In order to do this, \(F\) must be a real subfield, \(n\) must be strictly greater than \(2\), and \(E(n)^c\) (which has to be a root of unity in \(F\)) must be \(-1\). These facts can be checked directly, so this is faster than calculating the character table of the group and checking the value of a Frobenius-Schur indicator. The shortcut to calculate the local index of a cyclic cyclotomic algebra at an odd prime makes direct use of the following lemma of Janusz: If \(E_p/F_p\) is a Galois extension of \(p\)-local fields with ramification index \(e\), and \(z\) is a root of unity with order prime to \(p\), then \(z\) is a norm in \(E_p/F_p\) if and only if it is the \(e\)-th power of a root of unity in \(F\). ([Jan75], pg. 535). It follows that in order to calculate the local index at \(p\) of a cyclic cyclotomic algebra \([r,F,n,[a,b,c]]\), we first determine the splitting degree, residue degree, and ramification index \(e\) of the extension \(F(\zeta_n)/F\) at \(p\). Comparing the behaviour of the Galois automorphism \(\sigma_b\) to the behaviour of the Frobenius automorphism at \(p\) allows us to determine the order of the largest root of unity \(z\) with order coprime to \(p\) in the \(p\)-completion \(F_p\). The local index \(m_p\) is then the least power of \(E(n)^c\) that lies in the group generated by \(z^e\).
Calculation of the local index at the prime \(2\) makes use of the following consequence of ([Jan75], Theorem 5): A cyclic cyclotomic algebra \([r,F_2,n,[a,b,c]]\) over a \(2\)-local field \(F_2\) that is a subfield of a cyclotomic extension of the rational \(2\)-local field \(Q_2\) has Schur index at most \(2\). It has Schur index \(2\) if and only if \(4\) divides \(n\), \(F_2(\zeta_4)\) is totally ramified of degree 2, the Galois automorphism \(\sigma_b\) of \(F_2(\zeta_n)/F_2\) inverts all \(2\)-power roots of unity in \(F_2(\zeta_n)\), the order of \(E(n)^c\) is 2 times an odd number, and \((F_2:Q_2)\) is odd. The same approach to cyclotomic reciprocity makes it possible to check all of these conditions in the \(2\)-local situation.
The wedderga function that computes the \(p\)-local index of an ordinary irreducible character \(\chi\) of a finite non-nilpotent cyclic-by- abelian group \(G\) is based directly on a theorem of Benard [Ben76] that applies whenever the \(p\)-defect group of \(\chi\) is cyclic. We have to restrict our application of it to groups whose orders are small because the GAP records for irreducible Brauer characters are only available in these cases. In order to use this approach effectively, we developed a function that computes the defect group of the block containing a given ordinary irreducible character \(\chi\). This function makes use of the Min half of Brauer's Min-Max theorem (see Theorem 4.4 of [Nav98]), and thus is able to find the defect group directly from the ordinary character table. It is thus available for nonsolvable groups, even in cases where GAP's Brauer character records are not available. We are indebted to Michael Geline and Friederich Ladisch for discussions concerning the calculation of defect groups in GAP. The current algorithm we use is based on an approach suggested by Ladisch.
Some users may find it desirable to have an alternative description for the components of the Wedderburn decomposition of a group ring as algebras with structure constants, because the operations for algebras in GAP are designed for algebras with structure constants. We have provided such an algorithm that converts the output of WedderburnDecompositionInfo (2.1-2) into algebras with structure constants. Matrix rings over fields are converted directly. For components that are cyclotomic algebras, it calculates their defining group and defining character using those Wedderga operations, then uses IrreducibleRepresentationsDixon (Reference: IrreducibleRepresentationsDixon) to obtain matrix generators of an algebra isomorphic to the simple component corresponding to the character over a suitable field. An algebra with structure constants version of this is finally obtained by applying IsomorphismSCAlgebra (Reference: IsomorphismSCAlgebra w.r.t. a given basis) to this algebra.
When \(R\) is a semisimple ring, then every left ideal \(L\) of \(R\) is of the form \(L=Re\), where \(e\) is an idempotent of \(R\). Therefore, we can use the idempotents to characterize the decompositions of semisimple rings as a direct sum of minimal left ideals. In particular, let \(R=\oplus_{i=1}^t L_i\) be a decomposition of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family \(\{e_1,\dots,e_t\}\) of elements of \(R\) such that: each \(e_i\neq 0\) is an idempotent element, if \(i\neq j\), then \(e_ie_j=0\), \(1=e_1+\cdots+e_t\) and each \(e_i\) cannot be written as \(e_i=e_i'+e_i''\), where \(e_i',e_i''\) are idempotents such that \(e_i',e_i''\neq 0\) and \(e_i'e_i''=0\), \(1\leq i\leq \). Conversely, if there exists a family of idempotents \(\{e_1,\dots,e_t\}\) satisfying the previous conditions, then the left ideals \(L_i=Re_i\) are minimal and \(R=\oplus_{i=1}^t L_i\). Such a set of idempotents is called a complete set of orthogonal primitive idempotents of the ring \(R\). Such a set is not uniquely determined.
Let \(\mathbb F\) be a finite field and \(G\) a finite nilpotent group such that \(\mathbb F G\) is semisimple. Let \((H,K)\) be a strong Shoda pair of \(G\), \(C\in\mathcal{C}(H/K)\) and set \(e_C=e_C(G,H,K)\), \(\varepsilon_C=\varepsilon_C(H,K)\), \(H/K=\langle\overline{a}\rangle\), \(E=E_G(H/K)\). Let \(E_2/K\) and \(H_2/K=\langle\overline{a_2}\rangle\) (respectively \(E_{2'}/K\) and \(H_{2'}/K=\langle\overline{a_{2'}}\rangle\)) denote the 2-parts (respectively 2'-parts) of \(E/K\) and \(H/K\) respectively. Then \(\langle\overline{a_{2'}}\rangle\) has a cyclic complement \(\langle\overline{b_{2'}}\rangle\) in \(E_{2'}/K\). Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra \(F G\) (9.19), a complete set of orthogonal primitive idempotents of \(\mathbb F Ge_C\) is described (see [OVG11]) as the set of conjugates of \(\beta_{e_C}=\widetilde{b_{2'}}\beta_2\varepsilon_C\) by the elements of \(T_{e_C}=T_{2'}T_2T_E\), where \(T_{2'}=\{1,a_{2'},a_{2'}^2,\dots,a_{2'}^{[E_{2'}:H_{2'}]-1}\}\), \(T_E\) denotes a right transversal of \(E\) in \(G\) and \(\beta_2\) and \(T_2\) are given according to the cases below.
If \(H_2/K\) has a complement \(M_2/K\) in \(E_2/K\) then \(\beta_2=\widetilde{M_2}\). Moreover, if \(M_2/K\) is cyclic, then there exists \(b_2\in E_2\) such that \(E_2/K\) is given by the following presentation
\[\langle \overline{a_2},\overline{b_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=\overline{b_2}\hspace{1pt}^{2^k}=1, \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r \rangle,\]
and if \(M_2/K\) is not cyclic, then there exist \(b_2,c_2\in E_2\) such that \(E_2/K\) is given by the following presentation
\[\langle \overline{a_2},\overline{b_2},\overline{c_2}\mid \overline{a_2}\hspace{1pt}^{2^n}= \overline{b_2}\hspace{1pt}^{2^k}=\overline{c_2}\hspace{1pt}^2=1, \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r, \overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1}, [\overline{b_2},\overline{c_2}]=1 \rangle,\]
with \(r \equiv 1 {\rm \,mod\,} 4\) (or equivalently \(\overline{a_2}\hspace{1pt}^{2^{n-2}}\) is central in \(E_2/K\)). Then
\(T_2=\{1,a_2,a_2^2,\dots, a_2^{2^k-1}\}\), if \(\overline{a_2}\hspace{1pt}^{2^{n-2}}\) is central in \(E_2/K\) (unless \(n\leq 1\)) and \(M_2/K\) is cyclic; and
\(T_2=\{1,a_2,a_2^2,\dots,a_2^{d/2-1},a_2^{2^{n-2}},a_2^{2^{n-2}+1},\dots,a_2^{2^{n-2}+d/2-1}\}\), where \(d=[E_2:H_2]\), otherwise.
If \(H_2/K\) has no complement in \(E_2/K\), then there exist \(b_2,c_2\in E_2\) such that \(E_2/K\) is given by the following presentation
\[ \langle \overline{a_2},\overline{b_2},\overline{c_2}\mid \overline{a_2}\hspace{1pt}^{2^n}= \overline{b_2}\hspace{1pt}^{2^k}=1, \overline{c_2}\hspace{1pt}^2=\overline{a_2}\hspace{1pt}^{2^{n-1}}, \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,\\ \overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1},[\overline{b_2},\overline{c_2}]=1 \rangle, \]
with \(r\equiv 1 {\rm \,mod\,} 4\). In this case, \(\beta_2=\widetilde{b_2}\frac{1+xa_2^{2^{n-2}}+ya_2^{2^{n-2}}c_2}{2}\) and
\[T_2=\{1,a_2,a_2^2,\dots, a_2^{2^k-1},c_2,c_2a_2,c_2a_2^2,\dots,c_2a_2^{2^k-1}\},\]
with \(x,y\in\mathbb F\), satisfying \(x^2+y^2=-1\) and \(y\neq 0\).
When \(G\) is not nilpotent, we can still use the following description in some specific cases. Let \(G\) be a finite group and \(\mathbb F\) a finite field of order \(s\) such that \(s\) is coprime to the order of \(G\). Let \((H,K)\) be a strong Shoda pair of \(G\) such that \(\tau(gH,g'H)=1\) for all \(g,g'\in E=E_G(H/K)\), and let \(C\in\mathcal{C}(H/K)\). Let \(\varepsilon=\varepsilon_C(H,K)\) and \(e=e_C(G,H,K)\) (9.19). Let \(w\) be a normal element of \(\mathbb F_{s^o}/\mathbb F_{s^{o/[E:H]}}\) (with \(o\) the multiplicative order of \(s\) modulo \([H:K]\)) and \(B\) the normal basis determined by \(w\). Let \(\psi\) be the isomorphism between \(\mathbb F E \varepsilon\) and the matrix algebra \(M_{[E:H]}(\mathbb F_{s^{o/[E:H]}})\) with respect to the basis \(B\) as stated in Corollary 29.8 in [Rei03]. Let \(P,A\in M_{[E:H]}(\mathbb F_{s^{o/[E:H]}})\) be the matrices
\[ P= \left( \begin{array}{rrrrrr} 1 & 1 & 1 & \cdots & 1 & 1\\ 1 & -1 & 0 & \cdots & 0 & 0\\ 1 & 0 & -1 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & 0 & 0 & \cdots & -1 & 0\\ 1 & 0 & 0 & \cdots & 0 & -1\\ \end{array} \right) \quad {\rm and } \quad A= \left( \begin{array}{ccccc} 0 & 0 & \cdots & 0 & 1\\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & \cdots & 1 & 0\\ \end{array} \right). \]
Then
\[ \{x\widehat{T_1}\varepsilon x^{-1} \mid x\in T_2\langle{x_e}\rangle\} \]
is a complete set of orthogonal primitive idempotents of \(\mathbb F G e\) where \(x_e=\psi^{-1}(PAP^{-1})\), \(T_1\) is a transversal of \(H\) in \(E\) and \(T_2\) is a right transversal of \(E\) in \(G\) ([OVGnt]). By \(\widehat{T_1}\) we denote the element \(\frac{1}{|T_1|}\sum_{t\in T_1}{t}\) in \(\mathbb F G\).
A linear code of length \(n\) and rank \(k\) is a linear subspace \(C\) with dimension \(k\) of the vector space \(\mathbb F_q^n\). The standard basis of \(\mathbb F_q^n\) is denoted by \(E=\{e_1,...,e_n\}\). The vectors in \(C\) are called codewords, the size of a code is the number of codewords and equals \(q^k\). The distance of a code is the minimum distance between distinct codewords, i.e. the number of elements in which they differ.
For any group \(G\), we denote by \(\mathbb F_q G\) the group algebra over \(G\) with coefficients in \(\mathbb F_q\). If \(G\) is a group of order \(n\) and \(C\subseteq \mathbb F_q^n\) is a linear code, then we say that \(C\) is a left \(G\)-code (respectively a \(G\)-code) if there is a bijection \(\phi:E\rightarrow G\) such that the linear extension of \(\phi\) to an isomorphism \(\phi:\mathbb F_q^n\rightarrow \mathbb F_q G\) maps \(C\) to a left ideal (respectively a two-sided ideal) of \(\mathbb F_q G\). A left group code (respectively a group code) is a linear code which is a left \(G\)-code (respectively a \(G\)-code) for some group \(G\).
Since left ideals in \(\mathbb F_q G\) are generated by idempotents, there is a one-one relation between (sums of) primitive idempotents of \(\mathbb F_q G\) and left \(G\)-codes over \(\mathbb F_q\).
Note that each element \(c\) in \(\mathbb F_q G\) is of the form \(c=\sum_{i=1}^n f_i g_i\), where we fix an ordering \(\{g_1,g_2,...,g_n \}\) of the group elements of \(G\) and \(f_i\in \mathbb F_q\). If one looks at \(c\) as a codeword, one writes \([f_1 f_2 ... f_n]\).
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